\documentclass{article}

\usepackage{amsmath}
\usepackage{fixmath}

\renewcommand{\vec}[1]{\mathbold{#1}}
\newcommand{\mat}[1]{\mathbold{#1}}

\newcommand{\vx}{\vec{x}}

\newcommand{\vGamma}{\mat{\Gamma}}
\newcommand{\vX}{\mat{X}}

\begin{document}

We suggest the following notations:
\begin{itemize}
\item variable \(x\) 
\item vector \(\vx$ with \(i\)th element \(x_i\)
\item matrix \(\vX$ with \(i\)th column \(\vx_i\) and \((i,j)\)th element \(x_{ij}\)
\item random variable \(X\)
\end{itemize}

Longer equations should not use \texttt{\$\$..\$\$}.
Use the \texttt{equation} or \texttt{align} modes instead:
\begin{equation*}
\vx_i^T \vx_j = \vGamma.
\end{equation*}

\end{document}
